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B+ Tree

B+ Tree. By Li Wen CS157B Professor: Sin-Min Lee What is a B+ Tree Searching Insertion Deletion. What is a B+ Tree. Definition and benefits of a B+Tree

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B+ Tree

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  1. B+ Tree By Li Wen CS157B Professor: Sin-Min Lee What is a B+ Tree Searching Insertion Deletion

  2. What is a B+ Tree • Definition and benefits of a B+Tree 1.Definition: A B+tree is a balanced tree in which every path from the root of the tree to a leaf is of the same length, and each nonleaf node of the tree has between [n/2] and [n] children, where n is fixed for a particular tree. It contains index pages and data pages. The capacity of a leaf has to be 50% or more. For example: if n = 4, then the key for each node is between 2 to 4. The index page will be 4 + 1 = 5. Example of a B+ tree with four keys (n = 4) looks like this:

  3. What is a B+ Tree

  4. What is a B+ Tree • Question: Is this a valid B+ Tree? C E A B C D E F G H

  5. What is a B+ Tree Answer: 1.Both tree in slide 3 and slide 4 are valid; how you store data in B+ Tree depend on your algorithm when it is implemented. 2.As long as the number of data in each leaf are balanced, it doesn’t matter how many data you stored in the leaves. For example: in the previous question, the n can be 3 or 4, but can not be 5 or more than 5.

  6. What is a B+ Tree • Benefit: Every data structure has it’s benefit to solve a particular problem over other data structures. The two main benefits of B+ tree are: • Based on it’s definition, it is easy to maintain it’s balance. For example: Do you have to check your B+ tree’s balance after you edit it? No, because all B+ trees are inherently balanced, which make it easy for us to manipulate the data.

  7. What is a B+ Tree 2.The searching time in a B+ tree is much shorter than most of other kinds of trees. For example: To search a data in one million key-values, a balanced binary requires about 20 block reads, in contrast only 4 block reads is required in B+ Tree. (The formula to calculate searching time can be found in the book. Page 492-493)

  8. Searching • Since no structure change in a B+ tree during a searching process, so just compare the key value with the data in the tree, then give the result back. For example: find the value 45, and 15 in below tree.

  9. Searching • Result: 1. For the value of 45, not found. 2. For the value of 15, return the position where the pointer located.

  10. Insertion • Since insert a value into a B+ tree may cause the tree unbalance, so rearrange the tree if needed. • Example #1: insert 28 into the below tree. 25 28 30 Dose not violates the 50% rule

  11. Insertion • Result:

  12. Insertion • Example #2: insert 70 into below tree

  13. Insertion • Process: split the tree 50 55 60 65 70 Violate the 50% rule, split the leaf 50 55 60 65 70

  14. Insertion • Result: chose the middle key 60, and place it in the index page between 50 and 75.

  15. Insertion The insert algorithm for B+ Tree

  16. Insertion • Exercise: add a key value 95 to the below tree. 75 80 85 90 95 Violate the 50% rule, split the leaf. 75 80 85 90 95 25 50 60 75 85

  17. Insertion • Result: again put the middle key 60 to the index page and rearrange the tree.

  18. Deletion • Same as insertion, the tree has to be rebuild if the deletion result violate the rule of B+ tree. • Example #1: delete 70 from the tree This is OK. 60 65

  19. Deletion • Result:

  20. Deletion Example #2: delete 25 from below tree, but 25 appears in the index page. But… 28 30 This is OK.

  21. Deletion • Result: replace 28 in the index page. Add 28

  22. Deletion • Example #3: delete 60 from the below tree 65 Violet the 50% rule 50 55 65

  23. Deletion • Result: delete 60 from the index page and combine the rest of index pages.

  24. Deletion • Delete algorithm for B+ trees

  25. Conclusion • For a B+ Tree: • It is easy to maintain it’s balance. • The searching time is short than most of other types of trees.

  26. Reference • http://babbage.clarku.edu/~achou/cs160/B+Trees/B+Trees.htm • www.csee.umbc.edu/~pmundur/courses/CMSC461-05/ch12.ppt

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